MinT - Best Known (65−45, 65, s)-Nets in Base 128

Best Known (65−45, 65, s)-Nets in Base 128

(65−45, 65, 288)-Net over F₁₂₈ — Constructive and digital

Digital (20, 65, 288)-net over F₁₂₈, using

t-expansion [i] based on digital (9, 65, 288)-net over F₁₂₈, using
- net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
    - a function field by SÃ©mirat [i]

(65−45, 65, 386)-Net over F₁₂₈ — Digital

Digital (20, 65, 386)-net over F₁₂₈, using

t-expansion [i] based on digital (15, 65, 386)-net over F₁₂₈, using
- net from sequence [i] based on digital (15, 385)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 15 and N(F) ≥ 386, using
    - a curve from the manYPoints database [i]

(65−45, 65, 513)-Net in Base 128

(20, 65, 513)-net in base 128, using

t-expansion [i] based on (17, 65, 513)-net in base 128, using
- 7 times m-reduction [i] based on (17, 72, 513)-net in base 128, using
  - base change [i] based on digital (8, 63, 513)-net over F₂₅₆, using
    - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
        K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(65−45, 65, 96175)-Net in Base 128 — Upper bound on s

There is no (20, 65, 96176)-net in base 128, because

1 times m-reduction [i] would yield (20, 64, 96176)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 726 847228 350492 146182 714290 224655 649441 025176 011419 153992 758021 541676 333081 135778 209975 216101 735624 316869 092072 420742 732597 590631 943274 > 128⁶⁴ [i]